Quasilinear elliptic equations via perturbation method

Abstract

We present a new approach to studying a class of quasilinear problems including the so-called Modified Nonlinear Schrödinger Equations (MNLS). We show that solutions of quasilinear equations can be obtained as limits of 4-Laplacian perturbations.

Ground state solutions for a modified fractional Schrödinger equation with critical exponent

“…On the other hand, (23) implies that J ,q ( (t 0 )) < m ,q , a contradiction. This shows that u is a weak solution of the problem (7).…”

“…Equation 6 has been extensively studied in recent years, for example, see literature. [15][16][17][18][19][20][21][22][23][24][25][26][27][28] But, as 0 < < 1, a few results are seen for Equation (1) (see Li and Wu 29 ).…”

Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on RN

“…(1.1), such as, the existence of a positive ground state solution has been proved in [7,8] by using a constrained minimization argument; the problem is transformed to a semilinear one in [9][10][11] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [12,13]; perturbation method is also used to get the existence results on the bounded smooth domain in [14] and on the whole space in [15,16].…”

Multiple solutions for a modified Kirchhoff‐type equation in RN

“…Several methods can be used to solve the equation (1.3), such as, the existence of a positive ground state solution has been studied in [15,16] by using a constrained minimization argument; the problem is transformed to a semilinear one in [17][18][19] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [20,21]. Especially, in [22], the following problem: was studied via a perturbation method, where R N is a bounded smooth domian. In this paper, our aim is to search the existence of positive solutions, negative solutions, and sequence of high energy solutions for the whole space problem (1.1) via the perturbation method.…”

“…We have the following facts, their proofs see also [22], essentially. In order to the completeness, we give also their proofs.…”